ISOMORPHS AND NORMALIZATION

Any Sudoku can be scrambled into many so-called ISOMORPHS. But it is held that these isomorphs are mathematically equivalent, and the method of solution is identical for all of them.

But that only holds true if we ignore Symmetry Techniques (ST), such as introduced and developed by gurth, udosuk and RW. Scrambling can destroy or create symmetry, and so radically affect the solving process, at least for the human who relies on a CLEARLY APPARENT symmetry in order to use ST effectively.

So in effect isomorphs are NOT equivalent if they disturb or create symmetry.

In order to use ST effectively, it is essential that the Sudoku be scrambled into a symmetrical form, if it is not already symmetrical and if indeed a symmetry can be created by scrambling.

By NORMALIZATION is meant the scrambling of any asymmetrical Sudoku into a symmetrical version, where that is possible. Where it IS possible, there will usually be more than one symmetrical possibility. But it makes no mathematical difference which one we select: all the symmetrical isomorphs will be in a TRUE sense isomorphs.

What I am saying is that it is in future necessary for all serious solvers, including solver programs, to carry out the normalization procedure in order to reveal any latent symmetry and make ST accessible.

If this is not done on a regular basis, then many simple ST techniques will be missed, and the rating of the puzzle will be inflated unrealistically.

The actual process of nornmalization is quite easy to carry out by hand, it is interesting work and not too tedious, so solvers should be happy to have this new toy foisted on them.

You start by figuring out the symmetry by reason, then you interchange a few rows and columns, and that is all there is to it. When you have finished the puzzle, you change it back to its original asymmetic form. (Because YOUR symmetrical isomorph might not be the same as someone else's!)

For examples and practice, refer to the ongoing CHALLENGE on the thread Gurth's Puzzles on the General/Puzzle forum.

SYMMETRY TECHNIQUES

There are two types of symmetry that lend themselves to the development of ST (Symmetry Techniques). I have accustomed the world to many Emeralds, both variant and non-variant, based on 180-degree rotational symmetry, and solving techniques have been developed for these, notably by udosuk and RW.

But Emeralds are not necessarily confined to rotational symmetry only. An Emerald is strictly defined as ANY Sudoku having a symmetrical SOLUTION. If the clues are also symmetrical, then the Emerald is non-variant; if the clues are NOT symmetrical, but rotational symmetry is a GIVEN, then the Emerald is variant.

So, what other types of symmetry are there? In the context of ST, none that I have seen so far.

Except for the SECOND type of symmetry: DIAGONAL SYMMETRY.

I have seen no examples of DIAGONAL SYMMETRY except in my own Sudokus, commencing with GC28. Apparently this is something new in the world, or if not so, would appear to have been well and truly forgotten.

I consider these new emeralds with DIAGONAL SYMMETRY to be more important than the older ones with rotational symmetry: there are more useful techniques that can be applied to non-variant examples, also the puzzles are more satisfactorily constructed, and are likely to become more plentiful and popular than the older symmetry.

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